A simple pro edure to improve the pressureevaluation in hydrodynami ontext using theSPHDiego Molteni a& Andrea Colagrossi b,∗,aDipartimento di Fisi a e Te nologie Relative Universitá di Palermo,Viale delle S ienze, 90128 Palermo, Italy

bINSEAN, The Italian Ship Model Basin, Via di Vallerano 139, 00128 Roma,ItalyCESOS: Centre of Ex ellen e for Ship and O ean Stru tures, NTNU, Trondheim,NorwayAbstra tIn literature, it is well know that the Smoothed Parti le Hydrodynami s method anbe ae ted by numeri al noise on the pressure eld when dealing with liquids. This an be highly dangerous when an SPH ode is dynami ally oupled with a stru turalsolver. In this work a simple pro edure is proposed to improve the omputation ofthe pressure distribution in the dynami s of liquids. Su h a pro edure is based onthe use of a density diusion term in the equation for the mass onservation. Thisdiusion is a pure numeri al ee t, similar to the well known arti ial vis osityoriginally proposed in SPH method to smooth out the sho k dis ontinuities. As thearti ial vis osity, the density diusion used here goes to zero in reasing the numberof parti les re overing onsisten y and onvergen e of the nal numeri al s hemeadopted. Dierent arti ial density diusion formulas have been studied, payingattention to prevent unphysi al hanges of the ows. To show the improvementsof the new s heme proposed here, a suitable set of examples, for whi h referen esolutions or experimental data are available, has been tested.K ey words: Smoothed Parti le Hydrodynami s,, Weak- ompressibility, Freesurfa e ows, SPH pressure evaluation, Fluid-stru ture impa t problems,Convergen e tests∗ Corresponding author: Tel.: +39 06 50 299 343; Fax : +39 06 50 70 619.Email addresses: molteniunipa.it (Diego Molteni),a. olagrossiinsean.it (Andrea Colagrossi).Preprint submitted to Elsevier S ien e 17 January 2009

1 Introdu tionThe Smoothed Parti le Hydrodynami s (SPH) method is well suited for simu-lating omplex uid dynami s and is frequently used when the problem showsstrong free surfa e dynami s. Indeed in this ase a mesh free method hasto be preferred. However in many problems the attention is fo used on theuid kinemati s instead of the predi tion of the pressure eld. In general theow speeds look quite good, but he king the distribution of pressure, thesituation is dierent: large random pressure os illations are present due tonumeri al high frequen ies a ousti signal (see e.g. [2). In the simulations ofviolent liquid-solid impa ts the pressure value is very important and its eval-uation turns out to be riti al for pra ti al appli ations. Su h kind of impa tproblems are strongly time dependent and the analyti al pressure solution,to ompare with omputer pressure values, are omplex and limited to verysimple geometries (see e.g. [8). The result is that this problem is not deeplyinvestigated in the SPH literature.It is possible to lter the pressure time histories with a post-pro essor. This an be insu ient when the SPH solution for the uid is oupled at ea h timestep with the solid dynami s. Nonetheless, with the orre tion proposed inthis paper, the numeri al high-frequen y noise in the pressure eld is auto-mati ally ltered. In [3 the authors suggested a ltering of the density witha MLS integral interpolation as a way to redu e the numeri al noise in thepressure evaluation. That pro edure gives good results, but for long time sim-ulation it does not properly onserves the total volume of the parti le' systemsin e the hydrostati omponent has been improperly ltered (see e.g. [24).Following the idea of the smoothing pro ess, it is rather natural to think to adiusion pro ess. Furthermore it is well known that, in the framework of SPH,the diusion algorithm is onservative and therefore it preserves the mass on-servation. Obviously the diusion oe ient has to vanish as the numeri ala ura y in reases in order to avoid unphysi al ee ts and to re over the on-sisten y of the dis rete equations. We analyzed few simple problems and found lear eviden e of the dis repan y between analyti al and omputer pressurevalues and lear eviden e of the improvements. Parti ular attention has alsobe paid on the time integration s heme adopted whi h play an important roleboth in pressure evaluation as well as in the omputation osts.

2

2 Governing equations2.1 Field equationsIn free-surfa e SPH, the uid is generally onsidered invis id and the ow isfree to have rotational motion; the problem is thus governed by the Eulerequation in the domain Ω, whi h in the presen e of a generi external for eeld f reads:Du

Dt= −∇p

ρ+ f (1)in Lagrangian formalism

Dx

Dt= u (2)where x,u,p and ρ are respe tively the position of a generi material point,its velo ity, pressure and density.In the standard SPH formulation liquids are treated as weakly- ompressiblemedia, through an equation of state whi h dire tly link the density eld tothe pressure eld, i.e. :

p = c20(ρ− ρ0) (3)where c0 is the speed of sound evaluated in absen e of ompression, i.e. withρ = ρ0. Dierent hoi es an be made for the latter state equation (the more omplex Tait's equation is often used), generally with a very weak inuen eon the results (see e.g. [14). The speed of sound c =

√

dp/dρ must be at leastone order of magnitude greater than the maximum ow velo ityc > 10 max(|u|)Ω (4)This ondition ensures the density u tuations to remain lower than 1%ρ0. Inpra ti e a value of c smaller than the real one is adopted to avoid time stepstoo small in the time integration s heme but still satisfying the inequalityin (4). Conversely to the in ompressible assumption, the weakly- ompressibleapproa h leads to a set of equations whi h are not ellipti and therefore thenumeri al s heme an be written in an expli it way. From a pra ti al pointof view this leads to an algorithm whi h is easy to implement and an beparallelized in a very e ient way. On the other hand, when using the weakly ompressible approa h, numeri al pressure noise an easily develop, moreoverin region with negative pressure the so- alled tensile instability an be ex ited.(see [25, [5, [2).Further, sin e a liquid phase an generally be assumed as isentropi , the prob-lem is fully solved just ta king into a ount the ontinuity equation for the3

spe i volume eld vv =

1

ρ;

D[log(v)]

Dt= div(u) (5)The evolution of the spe i internal energy of the system is given by

ρDe

Dt= −p divu (6)However, due to the isentropi assumption the latter equation is de oupledfrom the other governing equations.

Boundary onditions. The uid boundary ∂Ω is omposed of a free surfa e∂ΩF and of solid boundaries ∂ΩB . On the free surfa e, two onditions mustbe veried. The kinemati ondition implies that the uid parti les initiallyon ∂ΩF will remain on the boundary As no surfa e tension is taken intoa ount, the dynami ondition states that the pressure is ontinuous a ross∂ΩF , therefore equal to the external pressure pe present on the other side.When pe is onstant, a trivial hange of pressure referen e leads to p = 0 onthe free surfa e, whi h is ommonly used by SPH pra titioners. These two ondition are impli itly veried in the SPH formalism (for details see [11).On solid boundaries ∂ΩB , a free-slip ondition is lassi ally assumed in free-surfa e SPH. A way to enfor e this ondition, is to use a lo al mirroring ofthe ow on the other side of the solid boundary. Namely, at ea h time-stepan image of the ow is generated on the other side of the solid boundary. Thethi kness of image ow is equal to the parti les' radius of intera tion. Moreoverto exa tly re over a free-slip ondition on the boundary, physi al propertieshave to be mirrored as well (see [3). The e ien y of this `ghost' approa h hasbeen underlined by Monaghan in its re ent review of the SPH method [21.The use of this `ghost' te hnique has been extensively validated both in termsof ow kinemati s and dynami s. Comparisons to other numeri al models onvarious appli ations have proved its e ien y (see e.g. [4, [10, [3, [2).3 SPH dis rete formulationIn SPH method, the uid domain Ω is dis retized as a nite number N ofparti les whi h represent small volumes of uid dV , ea h one with its ownlo al mass m and other physi al properties. In this ontext the divergen e ofthe velo ity eld for the generi parti le i an be expressed by the onvolutionsum:

〈divu〉i =∑

j

(uj − ui) · ∇Wj(xi) dVj (7)4

where uj and ui are respe tively the velo ity of the parti le j and i. Thesum is evaluated on the j neighbour parti le of the parti le i and the velo itydieren e (uj − ui) is weighted through a kernel fun tion W entred on theparti le j and evaluated onto the parti le i. In pra ti al SPH omputations,the hoi e of the kernel fun tion ae ts both the CPU requirements and thestability properties of the algorithm. In this work a renormalized Gaussiankernel has been adopted:Wj(xi) = W (r) =

e−(r/h)2 − C0

2π C1if r ≤ δ

0 otherwiseC0 = e−(δ/h)2 ; C1 =

∫ δ

0s[

(e−(s/h)2 − e−(δ/h)2]

ds

(8)where r = ‖xj − xi‖ is the Eu lidean distan e between the two parti les. Tomake its support ompa t a ut-o radius δ is introdu ed, typi ally set equalto 3h as for the lassi al fth-order B-spline support [16, h is alled smoothinglength and when it goes to zero the kernel fun tion W be omes a delta Dira fun tion. 1 .Using the interpolation formula (7) for the velo ity divergen e the ontinuityequation (5) an be dis retized as:

[

D log(v)

Dt

]

i

=∑

j

(uj − ui) · ∇Wj(xi) dVj ; ρi =1

vi(9)In [1 Bonet & Lok, have been demonstrated through a variational approa hthat if the ontinuity equation is expressed using the divergen e formula (7)the internal for e due to the pressure eld should be evaluated by:

T i =∑

j

(pj + pi)∇Wj(xi) dVj dVi (10)1 This hoi e of kernel has been motivated by the fa t that from a numeri al pointof view the behaviour of the renormalized Gaussian kernel is almost identi al to the lassi al Gaussian kernel (the maximum error between the two kernels is less than4 10−4). For what on erns the latter one the following properties are well estab-lished: (i) among ten tested kernel shapes, the Gaussian kernel appears to give thebest numeri al a ura y in the stable eld [9; (ii) the omparison of the Gaussiankernel to lassi ally used spline kernels showed that the former leads to better sta-bility properties [23; (iii) it presents also a lower omputational ost with respe t toevolved forms of spline kernels [2; nally (iv), its gradient an be straightforwardlyobtained from the evaluation of W itself.5

and onsequently the motion of ea h parti le an be integrated from the par-ti le a eleration whi h is given by Newton's se ond law:mi

[

Du

Dt

]

i= mif i − T i (11)

Summarizing, following the parti les in their Lagrangian evolution, the prob-lem governing equations (1), (2) and (5) are thus dis retized as

[

D log(v)

Dt

]

i

=∑

j

(uj − ui) · ∇Wj(xi) dVj

[

Du

Dt

]

i= − vi

∑

j

(pj + pi)∇Wj(xi) dVj + f i

[

Dx

Dt

]

i= ui

dVi = mi vi ; ρi =1

vi

; pi = c20(ρi − ρ0)

(12)Using the Bonet & Lok pro edure those equations for the parti le systempreserve linear and angular momentum. To evolve in time this problem (12),dierent temporal s heme an be used and we dis uss this aspe t in the nextse tions. The temporal dis retization is dynami ally linked to the spatial oneby satisfying the stability onstraints due to the lo al speed of sound, and tothe lo al value of the parti le velo ity, a ording to [18. Starting from an initialdistribution (xi) of parti les with given masses mi (whi h remain onstant intime), densities ρi and velo ities ui, the PDEs (12) are solved at ea h time-step at the parti le lo ations, providing the unknown time derivatives of theparti le positions, velo ities and densities; the pressure and the density beingdire tly linked by the state equation of (3).The hara teristi SPH dis retization parameters are: (i) the ratio h/L, whereL is a typi al length s ale of the problem, and (ii) the number N of parti leswithin the intera tion radius. Roughly speaking, the ratio d = h/N is theequivalent of the grid spa ing in mesh-based methods. The resulting totalnumber of parti les N depends on the appli ation onsidered, in all the asesstudied in this paper the sele ted number of neighbours N is about 50 in 2D, orresponding to the ratio h/d = 1·33. The onvergen e is studied by varyingthe initial distan e between the parti les.6

4 Introdu tion of diusive terms Arti ial vis osity It is well know in literature that the SPH equations(12) are not stable when dealing with liquids and annot be used for pra ti alproblems, unless a small arti ial vis ous term of the form

V i = α h c0∑

j

πij∇Wj(xi) dVj (13)is added in the dis retized momentum equation of (12). In the present work,the expression proposed by Monaghan [16 is rearranged as:πij = min

[

(ui − uj) · (xi − xj)

|xi − xj |2; 0

]

, ∀i 6= j (14)The equivalent kinemati vis osity asso iated with V i has the form 15/112αc0h(see [15,[21), its ee t tends therefore to zero with the neness of dis retiza-tion. Therefore the onsisten y between the dis rete equations and the ontin-uous one is preserved. The value of α for a given spatial resolution h has to be hosen in way that the numeri al vis ous ee t annot introdu e unphysi alee ts. Arti ial density diusion The main target of this work is to show thatthe use of the standard arti ial vis osity (13) for hydrodynami problems ould not be su ient to prevent the development of instabilities as well asthe presen e of high frequen ies numeri al noise on the pressure eld. Instead,the addition of a diusive term in the ontinuity equation an avoid thesedrawba ks giving more reliable results. We propose a density diusion termof this type

Di = ξ h c0∑

j

ψij · ∇Wj(xi) dVj (15)whi h has a similar mathemati al stru ture of the Monaghan's arti ial vis- osity. The ψij is written as:ψij = 2

(

vi

vj− 1

)

(xi − xj)

|xi − xj |2∀ i 6= j (16)As mentioned before for the arti ial vis osity also this se ond orre tionvanishes in reasing the spatial resolution (de reasing h), re overing the on-sisten y with the equation (5).

The new SPH s heme proposed Summarizing the rst two equations of7

the parti le' evolution (12) are rewritten in the form:

[

D log(v)

Dt

]

i

=∑

j

(uj − ui) · ∇Wj(xi) dVj + ξ h c0∑

j ψij · ∇Wj(xi) dVj

[

Du

Dt

]

i= − vi

∑

j

(pj + pi)∇Wj(xi) dVj +αh c0∑

j πij∇Wj(xi) dVj + f i(17)The two parameters [ξ, α] give the intensity of the numeri al orre tions in-trodu ed. For the appli ations investigated in this work these two parametersare usually lower that 0.1 this means that the extra ost of omputationaltime is not relevant sin e: (i) the time step is not largely ae ted by those orre tions, (ii) the omputation of the extra terms is kept down sin e someoperations required by the arti ial vis osity an be used for the evaluation ofthe density diusion. In the next se tion we show that with a orre t hoi eof [ξ, α] it is possible to improve the SPH results.In the following se tion the problems have been studied using dierent timeintegration s hemes for whi h the Courant number will be indi ated with CFLand the link with ∆t is expressed as:∆t = CFL min

i

(

h

c0(1 + ξ/4) + |ui| + σi

)

; σi = h maxj

∣

∣

∣

∣

∣

(ui − uj) · (xi − xj)

|xi − xj |2∣

∣

∣

∣

∣(18)where the minimum is over all parti les and the maximum is over the parti le'neighbours of the generi i-parti le. The CFL fa tor depends by the numeri als heme adopted.5 Results for test problemsIn this paper the results of the following problems will be dis ussed: a uxgenerated by jet impinging orthogonally on a solid plate, a tank of water loseto the hydrostati equilibrium and nally an unsteady impa t problem ausedby a dam break ow.5.1 Jet impinging on a at plateThe jet on plate problem is a steady state problem, for whi h an analyti alsolution for the pressure is available. A two dimensional water jet impinges ona rigid plate without any physi al vis osity. The wall denes the x axis. Theplate may be in lined or orthogonal to the impinging ow. This problem hasbeen solved analyti ally by Mi hell in 1890 in a very omplex way [13. Taylor8

-1.5 -1 -0.5 0

0

0.5

1

1.5

2

0.000000E+00

H/L=0.2

Inflow Section

Outflow Section -VREF

x

y

L

t U/L =

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

t U/L

VREF(t)/U

-1.5 -1 -0.5 0

0

0.5

1

1.5

2

0.50.450.40.350.30.250.20.150.10.050

0.100000E+02

H/L=0.2

Inflow Section

Outflow Section

x

y

L

t U/L =

P/ρU2

PressureProbe p0

Fig. 1. Left: Initial onditions and set up of the parti les for the 2D jet impinging anorthogonal plate. Right: Parti les' onguration at dimensionless time tU/L = 10(the steady state is pra ti ally rea hed). The parti les are olored with pressurelevels. The red lines indi ate the positions of the inow and outow se tions.[26 gives an impli it expression for the speed and the pressure at the wall andwe report here the spe i fun tion in the ase of orthogonal impingingx

H=

1

πlog

(

1 + q

1 − q

)

+ arcsin(q) ; q = − 2u

u2 + 1(19)where x is the horizontal distan e, the physi al referen e length is the halfwidth of the impinging jet whi h has been indi ated with H (see left plotin gure 1). u is the dimensionaless speed, the referen e speed is the inowvelo ity of the jet U . From the eld velo ity u obtained by eq. (19) the pressureis evaluated through the Bernoulli theorem.Left plot of gure (1) shows the initial set up of the parti les. Due to thesymmetry of the problem the simulation has been arried out only into halfdomain. At the inow (y = 1·5) the pressure is zero, the parti les are a eler-ated smoothly to the regime inow speed U to avoid the noise of an impulsivestart. At x = −1.5 an outow se tion is present. The domain of interest is

[−L, 0] × [0, L]; the inow and outow se tions are far enough to avoid anyinuen e related by the inje tion of new parti les and the removal of outgoingones. To solve the problem in the weakly ompressible regime the SPH speedof sound c0 (see eq. (3)) is set equal to 12U . The simulations were all arriedon with an arti ial vis osity α = 0.1. Of ourse a larger arti ial vis osity oe ient would produ e more stable results, but the resulting pressure andspeed proles would be ae ted by this larger unphysi al vis osity. To he kheuristi ally the onvergen e of the algorithm three dierent spatial resolu-tion have been adopted varying the initial distan e from the parti les dx (i.e.9

varying the number of the parti les N in the domain). On the right plot of thegure (1) a stable SPH solution is shown at dimensionless time tU/L equal to10 for whi h the steady state is rea hed. The parti les are oloured with thepressure values and the pressure p0 at the stagnation point has to be equal to0·5.5.1.1 Numeri al solution using a Modied Euler s heme.We start to solve the problem with a modied Euler s heme for the timeintegration. It onsists in a predi tion and a orre tor steps that uses anaverage of the rates of hange at the two time instants (see i.e. [20). Re-writing the evolution parti le equations as:J = log(1/ρ),

[

DJ

Dt

]

i= Ei ;

[

Du

Dt

]

i= F i ;

[

Dx

Dt

]

i= ui (20)with the modied Euler those equations evolve in time following the s heme:

Jn+1,∗i = Jn

i + ∆t Eni

un+1,∗i = un

i + ∆tF ni

xn+1,∗i = xn

i + ∆tuni +

∆t2

2F n

i

⇒

Jn+1i = Jn

i + ∆t(En

i + En+1,∗i )

2

un+1i = un

i + ∆t(F n

i + Fn+1,∗i )

2

xn+1i = x

n+1,∗i (21)where the supers ript n,(n+ 1, ∗) and (n+ 1) indi ate respe tively the a tualtime instant, the predi ted one and orre ted new time instants. Following the

0 1 2 3 4 5-0.5

0

0.5

1

1.5

2

2.5

tU/L

p0 /ρU2 H/d = 10[α;ξ] =[0.0;0.0]

[α;ξ] =[0.1;0.0]

[α;ξ] =[0.1;0.1]

Fig. 2. 2D jet impinging an orthogonal plate problem: time history of the SPHstagnation pressure evaluated without any numeri al orre tions (dotted line), witharti ial vis osity α = 0.1 (dashed line), with both arti ial vis osity and densitydiusion [α = 0.1, ξ = 0.1]. The initial distan e among the parti le is d = H/10.10

s heme proposed in [20 the positions of the parti les are not orre ted. Usinga CFL value equal to 0·3 this s heme is stable.Figure (2) shows the time history of the stagnation pressure using this times heme. Three dierent simulation have been arried out: (i) using no numer-i al orre tions (i.e. [α = 0, ξ = 0]), (ii) using the standard arti ial vis osity([α = 0.1, ξ = 0]) and (iii) adding also the density diusion ([α = 0.1, ξ = 0.1]).In the region where the jet is a elerated against the wall and after a transientphase the pressure starts to de rease and moves lose to the steady state valuep = 1/2 ρ0U

2. The rst solution present large amplitude pressure os illationswhile the se ond two solutions are more a eptable. In parti ular, using thedensity diusion orre tion (see paragraph 4) the SPH solution has smalleros illations loser to the analyti al one.After this rst analysis a onvergen e study has been performed. Three dier-ent simulations have been arried out hanging the spatial resolutions H/d =10, 20, 40, whereH is the semi-width of the jet and d the initial distan e amongthe parti les. Figure (3) shows the onvergen e of the time history of the stag-nation pressure. The plot on top is related to simulations where only arti ialvis osity has been used (i.e. [α = 0.1, ξ = 0]). In this ondition for t > 5L/Uthe numeri al solution onverges to the analyti al one, nonetheless during thetransitory phase the SPH solution seems to not onverge properly. In fa t thenumeri al solution drasti ally hange in reasing the spatial resolution fromH/d = 10 to H/d = 20 and looking the parti les distribution near to the timeinterval tL/U = (0·6, 1·2) unphysi al lumping of parti les appears, ausingnumeri al overpressure levels. Instead, if we use the density diusion (ξ = 0.1)the onvergen e is more uniform on the whole time interval, as it is shown onthe bottom plot of gure (3). Furthermore with the presen e of density diu-sion orre tion the parti les distribution during the impa t transitory phaseappears quite regular.5.1.2 Numeri al solution using a Modied Verlet s heme.Sin e re ently symple ti time integrators have been used in SPH [21, theproblem of the jet impinging on a at surfa e has been solved also with a Verletse ond order integrator. Sin e the symple ti time integrators are developedunder no dissipation assumption, the Verlet s heme has been slightly modied11

[22 as:

un+1/2i = un

i + ∆t/2 F ni

xn+1/2i = xn

i + ∆t/2 uni

Jn+1/2i = Jn

i + ∆t/2Eni

⇒

un+1i = un

i + ∆tFn+1/2i

⇓xn+1

i = xn+1/2i + ∆t/2 un+1

i

⇓Jn+1

i = Jn+1/2i + ∆t/2En+1

i

(22)

0 1 2 3 4 50

0.5

1

1.5

tU/L

p0 /ρU2

H/d = 40

H/d = 20

H/d = 10

[α;ξ] =[0.1;0.0] cfl=0.30

0 1 2 3 4 50

0.5

1

1.5

tU/L

p0 /ρU2

H/d = 40

H/d = 20

H/d = 10

[α;ξ] =[0.1;0.1] cfl=0.30

Fig. 3. 2D jet impinging an orthogonal plate problem: Convergen e analysis hang-ing the initial distan e among the parti le d = H/10 (solid line), d = H/20 (dashedline), d = H/40 (dashdot line). Top: time history of SPH stagnation pressure eval-uated with arti ial vis osity and without density diusion (i.e. [α = 0.1, ξ = 0.0]).Bottom: time history of SPH stagnation pressure evaluated with arti ial vis osityand density diusion (i.e. [α = 0.1, ξ = 0.1]).12

All the evolution variables are evaluated at the midpoint time instant n+1/2then the parti le' velo ities are evaluated at the new time instant, n+1. Thenthe velo ities are used to update the parti les' positions. Finally, using thenew velo ities and positions of the parti les the density rate of hange are al ulated to update the density eld at n+ 1 time level. With this numeri als heme it is possible to adopt a larger CFL fa tor. Indeed for the simulationshown here it has been set equal to 1·2. However one has to take into a ountthat the Verlet s heme requires one more intera tion y le over the parti le'system for the evaluation of the density eld.Figure (4) shows the onvergen e analysis of the stagnation pressure timehistory adopting this se ond time integrator. Those results are obtained usingonly the arti ial vis osity. The Verlet s heme gives a more reliable behaviourduring the transitory phase respe t to the Modied Euler s heme presentedin the paragraph (5.1.1). In parti ular in reasing the spatial resolution H/dthe unphysi al lumping observed with the Modied Euler s heme disappears.Adding the density diusion orre tion (i.e. ξ = 0.1) the same results dis ussedin the paragraph (5.1.1) are obtained. Again the use of the parameters [α =0.1; ξ = 0.1]) allows to obtain solutions hara terized by mu h less numeri alnoise on the pressure eld. This is quite evident looking the gure (5) whi hshows the pressure distribution evaluated with and without density diusion orre tion.Figure (6) shows the omparison of the pressure a ting on the solid plate eval-uated with dierent numeri al s heme and using or not the density diusion

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

tU/L

p0 /ρU2

H/d = 40

H/d = 20

H/d = 10

[α;ξ] =[0.1;0.0]

Fig. 4. 2D jet impinging an orthogonal plate problem: time history of SPH stag-nation pressure evaluated with arti ial vis osity and without any density diusion(i.e. [α = 0.1, ξ = 0.0]). Convergen e analysis for d = H/10 (solid line), d = H/20(dashed line), d = H/40 (dashdot line). The dis rete evolution equations are inte-grated in time with the modied Verlet s heme (see paragraph 5.1.2).13

orre tion. The numeri al solutions have been ompared with the analyti alone obtained by Taylor (19). The results presented in gure (6) are obtainedwith the nest spatial resolution H/d = 40 and interpolating the pressure par-ti les on the solid wall through a MLS integral interpolation (see i.e. [3, [2).It is lear from the plots presented in gure (6) that the SPH pressure evalu-ated with the density diusion orre tion are loser to the referen e analyti alsolution.5.2 Settling of a stratied uid in a partially lled tank.Two uids, horizontally stratied, are left to settle in a squared tank withside L. The heavier uid A (ρA = 1000Kg/m3) rests on the tank bottom, thelighter one B (ρB = 500Kg/m3) is on the top of it. The two uid layers have

-0.4 -0.2 0 0.2 0.4

0

0.2

0.4

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

x/L

y/L

ξ=0.1ξ=0.0

p/ρU2

Fig. 5. 2D jet impinging an orthogonal plate problem: enlarged view of the parti les'distribution lose to the stagnation point obtained without the density diusion orre tion (x-negative), and with diusion orre tion (x-positive). Dimensionalesspressure values are shown. These results have been obtained using the modiedVerlet s heme (see paragraph 5.1.2) with a CFL fa tor equal to 1·2.14

-1 -0.8 -0.6 -0.4 -0.2 00

0.1

0.2

0.3

0.4

0.5 Verlet,ξ=0.0

Verlet,ξ=0.1

Modified Euler ξ=0.1

Analytical Solution (G. Taylor 1960)

p/ρU2

x/L

SPH spatial resolution: H/d = 40Fig. 6. 2D jet impinging an orthogonal plate problem: with the modied Verlets heme (see paragraph 5.1.2).the same hight (HA = HB = H/2) (see gure 7). The velo ity eld is initiallyset to zero for both the two uids.Sin e the initial pressure distribution is zero for all the parti les, the systemevolves in time, os illating around its equilibrium onguration. During the omputation, the kineti energy generated by the initial non-equilibrium, willbe dissipated through the arti ial vis osity. After su h transitory phase, thekineti energy will vanish and the pressure eld will rea h the hydrostati distribution:

p(x, y) = ρB g (HA +HB − y) if y > HA

p(x, y) = ρB g HB + ρA g (HA − y) if y ≤ HA

(23)where y = HA +HB is the initial verti al oordinate of the at free surfa eThis test is very simple, but it shows whether any numeri al perturbing ee tis present. Indeed long time simulations show that, in some onditions, witha SPH solver the perturbations due to the resettling of the parti les initialpositions an grow indu ing an unphysi al haoti motion.The left plot of gure (7) shows the initial onguration with parti les po-sitioned on a regular artesian latti e and, therefore, ea h parti le has aninitial volume equal to dV0 = d × d. The speed of sounds are set equal toc0A = 12

√gL and c0B = 17

√gL respe tively for the uid A and B. The initialdistan e among parti les is d = H/16. The right plot of gure (7) shows anew stable parti le's onguration after 20000 intera tions. Sin e the numberof intera tions is quite high for this type of test, a 4th order Runge-Kutta15

-0.4 -0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

HB

L

t(g/L)½=0.0

x/L

y/L

HA

-0.4 -0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

t(g/L)½=100

x/L

y/L

Fig. 7. Settling of two dierent uids in a partially lled tank. Left: initial parti- le's onguration. The parti les are oloured a ording their density values (red:ρA = 1000Kg/m3; blue: ρB = 500Kg/m3) Right: a new stable parti le's ongura-tion rea hed at dimensionless time instant t

√

g/L = 100.

0 10000 20000 30000 40000

10-6

10-5

10-4

10-3

10-2

10-1

100

101

t(g/L)½=100

EKin/EKin-Max H/d = 32[α;ξ] =[0.03;0.0]

[α;ξ] =[0.03;0.1]

NIteractionsFig. 8. Settling of two dierent uids in a partially lled tank. Time histories ofthe kineti energy of the SPH parti les' system using only the arti ial vis osity(solid line), using both arti ial vis osity and density diusion numeri al orre tions(dashed line).s heme has been adopted 2 .Figure 8 shows the evolution of the parti le kineti energy obtained with aspatial resolution (d = H/32). For this problem α has been set equal to 0·03whi h is a value large enough to stabilize the simulation. During the rst timesteps the kineti energy starts to os illate due to the motion of the parti lesaround their initial position. Figure 8 shows also the time histories of the2 For the problems studied in this paper it was found that the results given by themodied Verlet s heme (see paragraph 5.1.2) are quite similar to those produ ed bythe 4th order Runge-Kutta s heme. 16

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0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4+

y/L

y=HA

[α=0.03;ξ=0.0]

[α=0.03;ξ=0.1]

Analytical Solution

P/ρAgL

t(g/L)½=100Fig. 9. Partially lled tank at rest ondition.kineti energy when the density diusion orre tion is applied (dashed line).It is quite evident that the density diusion helps to a elerate the dampingof the kineti energy.Figure 9 shows the pressure prole evaluated at dimensionless time t√g/L =100 without and with the density diusion orre tion. In the latter ase, thepressure prole is less noisy and loser to the analyti al solution (eq. 23). Swit hes on the density diusion orre tion.To obtain these results two swit hes have been applied to the density diusion orre tion. The rst one allows the density diusion only for the parti les be-longing to the same uid. This avoids the unphysi al density diusion amongdierent uids. The se ond swit h allows the diusion only when the verti alpressure gradient between two parti les is larger than the one required by thehydrostati gradient. Without this se ond swit h, the density diusion at-tens also the hydrostati pressure (eq. 23) in an enough long simulation. 3 .A tually the orre tion on the ontinuity equation proposed in this paper hasto be applied only to the dynami pressure omponent. A similar behaviourhas been found using the density re-initialization with a MLS interpolationproposed in [3. This instability has been noted in [24, where the author pro-posed this lter only on the dynami pressure eld leaving the hydrostati part unaltered.3 This kind of swit h has to be apply whenever a body for e is present.17

Summarising, the ψij fa tor of equation (16) has been hanged in:ψij =

0 if parti le i and j do not belong to the same uid or i = jotherwise

2(

vi

vj− 1

)

(xi − xj)

|xi − xj|2if |pj − pi|

ρi g |yi − yj|> 1

0 otherwise (24)In the next paragraph an impa t problem aused by a dam break ow will bestudied. In this ase both gravity as well as impa t events play an importantrole.5.3 Fluid-Stru ture and Fluid-Fluid impa ts generated by a Dam-Break Flow.The problem of a dam break with the subsequent impinging of the waterow on the opposite wall is a good test for highly dynami al problem. Figure(10) shows the initial ondition. The reserve of water has an initial heightH and a length set equal to L = 2H . A dam is positioned at longitudinalposition x = 2H and it is suddenly removed at initial time t = 0. Sin e theSPH method used here is based on a weakly ompressible assumption, it isnot possible to start the problem with an hydrostati pressure ondition (formore details see [4, [2). Therefore a Poisson equation for the pressure hasbeen solved for t = 0 whi h gives the proper ondition for the pressure and

0 1 2 3 4 5 6

0

1

2

3

10.90.80.70.60.50.40.30.20.10

x/H

y/HP/ρgH

P3

P1

P2

LW

L

H

Initial condition for thepressure field:∇ 2p (t=0) = 0p=0 on the free surface∂p/∂n = 0 on solid walls

Fig. 10. Impa ts generated by a dam-break ow: Sket h of the problem, adoptednomen lature and denition of the Poisson equation for the initial pressure ondi-tions. 18

onsequently for the initial density distribution. The solution of this problem an be derived in a semi-analyti way (see i.e. [8). For this problem the speedof sound c0 is set equal to c0 = 20√gH. A verti al wall is pla ed at distan e

(Lw − L) from the broken dam, and the uid owing along the initially dry-de k impa ts eventually against it (uid-stru ture impa t). After a run up - rundown y le, the water overturns ba kwards onto the underlying uid (uid-uid impa t). This evolution is shown in gure (11) and it has been omputedwith SPH using a Runge Kutta 4th order with a Courant number CFL = 2·5and without any density diusion orre tion (ξ = 0). The arti ial vis osity oe ient is set equal to α = 0.03 to have a numeri al Reynolds number largeenough to redu e the dissipation of me hani al energy (see for details [3).From the pressure ontour plots shown in gure (11) it is possible to see thatduring the impa t phases high unphysi al a ousti noise has generated. Tolimit su h numeri al pressure noise the density diusion orre tion has beenapplied and for this kind of appli ations the equation (16) has been hangedas:ψij = 2

(

vi

vj− 1

)

(xi − xj)

|xi − xj |2Wj(xi)

W (d/2)∀ i 6= j (25)The ratioWj(xi)/W (d/2) is here added to give a larger weight to loser neigh-bourhood parti les. In parti ular the density diusion orre tion in reaseswhen two parti les be ome loser than the half of their initial distan e d. Thelatter ondition an be realized only during violent impa t events or in regionof high deformations as in the ase dis ussed in this se tion.The swit hes on the density diusion introdu ed in paragraph 5.2 do not play

0 1 2 3 4 50

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.800934E+00

x/H

y/Ht(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.0]

0 1 2 3 4 50

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.400016E+01

x/H

y/Ht(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.0]

0 1 2 3 4 50

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.160062E+01

x/H

y/Ht(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.0]

0 1 2 3 4 50

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.560126E+01

x/H

y/Ht(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.0]

0 1 2 3 4 50

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.320145E+01

x/H

y/Ht(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.0]

0 1 2 3 4 50

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.640003E+01

x/H

y/Ht(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.0]

Fig. 11. Impa ts generated by a dam-break ow: time evolution al ulated with SPHusing the arti ial vis osity orre tion (α = 0.03) and without any density diusion(ξ = 0) The initial distan e between parti les is d = H/100. The ontour plots arerelative to the pressure eld. Time in reases from top to bottom and from left toright. 19

0 1 2 3 4 5

0

0.5

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.800624E+00

x/H

y/H

P1

P2

t(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.20]

0 1 2 3 4 5

0

0.5

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.400040E+01

x/H

y/H

P1

P2

t(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.20]

0 1 2 3 4 5

0

0.5

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.160002E+01

x/H

y/H

P1

P2

t(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.20]

0 1 2 3 4 5

0

0.5

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.560096E+01

x/H

y/H

P1

P2

t(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.20]

0 1 2 3 4 5

0

0.5

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.320011E+01

x/H

y/H

P1

P2

t(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.20]

0 1 2 3 4 5

0

0.5

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.645098E+01

x/H

y/H

P1

P2

t(g/H)½ =

x/H

y/HP/ρgH

P1

P2

H/d = 100[α ; ξ ] = [0.03 ; 0.20]

Fig. 12. Impa ts generated by a dam-break ow: time evolution al ulated withSPH using both the arti ial vis osity and density diusion orre tions (see eq. 25)[α = 0.03, ξ = 0.2. The initial distan e between parti les is d = H/100. The ontourplots are relative to the pressure eld. Time in reases from top to bottom and fromleft to right.a relevant role in this type of ow: only one uid is present and the simulationstops before density diusion ee ts be ome appre iable.The benets of the orre tion (25) are visible in gure (12) where more reliablepressure ontour plots are reported. In parti ular it is evident the redu tionof the pressure noise during the impa t stages.To quantify more in detail the role of the density diusion orre tion a om-parison with experimental data has been reported in gure (13). In this gure,the time histories of the numeri al predi tion of the pressure a ting on probeP1 (see 10) have been ompared with experimental data [12. The omparisonis limited to the time interval t√g/H ∈ (2, 5) when the impa t between thedam break ow and the right solid wall takes pla e 4 . Three dierent spa-tial resolutions (d = H/25 , H/50 , H = 100) have been adopted to ontrolthe onvergen e of the numeri al predi tions. On top plot of gure (13) theSPH results have been obtained using only the arti ial vis osity orre tion(α = 0.03, ξ = 0). In reasing the number of parti les adopted the SPH predi -tions onverge toward the experimental time series; nonetheless an high levelof numeri al noise is always present. The use of the density diusion orre tion(ξ = 0.2) an redu e drasti ally this unphysi al pressure noise as shown in thebottom plot of gure (13).4 For a mu h longer time omparison the presen e of the air phase annot be anymore negle ted as ommented in [3. 20

2 3 4 5-1

-0.5

0

0.5

1

1.5

2

2.5 P1/ρgH

t(g/H)½

H/d = 25

H/d = 50

H/d = 100

Exp. Data

[α ; ξ]=[0.03 ; 0.0]

2 3 4 5-1

-0.5

0

0.5

1

1.5

2

2.5 P1/ρgH

t(g/H)½

H/d = 25

H/d = 50

H/d = 100

Exp. Data

[α ; ξ]=[0.03 ; 0.2]Fig. 13. Impa ts generated by a dam-break ow: SPH pressure time evolution onprobe P1 (see gure 10) ompared with experimental time re ord [12. Three dierentspatial resolution have been adopted to ontrol the onvergen e of the numeri alpredi tions (d = H/25H/50H = 100). Top plot: Results obtained using only thearti ial vis osity orre tion (α = 0.03, ξ = 0). Bottom plot: Results obtained usingboth arti ial vis osity and density diusion orre tions [α = 0.03, ξ = 0.2] (seeequations (25 and se tion (4)).Finally a he k in time of the me hani al energy onservation has been done to ontrol that the new numeri al orre tions suggested in this paper do not intro-du e too mu h numeri al dissipation. Figure (14) shows the time evolution ofthe me hani al energy (kineti plus potential energies). The me hani al energyis made non-dimensional by the potential-energy imbalan e ∆E = E2 − E1 orresponding to the stati ongurations before the dam break (E1) and withthe same amount of uid uniformly distributed along the horizontal bottom(E2). It is quite visible that the impa t events ause a loss of energy due tothe arti ial vis osity a tion. Nonetheless this plot highlighted that the use of21

0 2 4 6 8 10-0.6

-0.4

-0.2

0

∆E = E2 - E1=ρgLH2(Lw-L) / 2LW

(EKin + E Pot ) / ∆E

t(g/H)½

[α ; ξ] = [0.03 ; 0.0]

[α ; ξ] = [0.03 ; 0.2]E2

E1

t=∞

t=0

Fig. 14. Impa ts generated by a dam-break ow: Time history of the me hani alenergy using only the arti ial vis osity orre tion (α = 0.03, ξ = 0, solid line) andusing both the arti ial vis osity and density diusion orre tions (α = 0.03, ξ = 0,dash dotted line). The me hani al energy is made dimensionless by the potential-en-ergy imbalan e ∆E = E2 − E1 dened in the pi ture.the density diusion produ es a better onservation of the total energy of thesystem.6 Con lusionThe SPH approa h to in ompressible uid dynami s through the weakly om-pressible approximation is attra tive due to its simpli ity of implementationand treatment. However, in general, the pressure values suer noise due tonumeri al u tuations. An averaging therapy on the pressure eld is somehowobvious and it has been proposed in literature (see i.e. [3). Unfortunately inthat ase also the hydrostati pressure omponent is ltered indu ing instabili-ties whi h are di ult to ontrol. In this work it is shown that the introdu tionof a proper diusive term in the ontinuity equation in reases the smoothnessand the a ura y of pressure proles. A simple expression of the diusion o-e ient is proposed. This is similar to the Monaghan arti ial vis osity termwith some orre tions whi h have to be added depending the nature of theproblem studied. It is also shown that this orre tive method does not alterthe mat h of the numeri al solution with the analyti al one, at least for the ases studied. Although a more general mathemati al investigation is required,we think it is worthwhile and useful to ommuni ate these promising results22

and to highlight the relevan e of them in the eld of violent free surfa e ows.A knowledgements This work was partially supported by the Centre forShips and O ean Stru tures (CeSOS), NTNU, Trondheim, within the "Vi-olent Water-Vessel Intera tions and Related Stru tural Load" proje t, andpartially done within the framework of the "Programma Ri er he INSEAN2007-2009" and "Programma di Ri er a sulla Si urezza" funded by MinisteroInfrastrutture e Trasporti.A Theoreti al onsiderations on pressure and velo ity numeri alos illationsIn the ase studied in se tion 5.1 for zero arti ial density diusion, it hasbeen shown that the pressure eld is hara terized by large numeri al u tua-tions. Figure A.1 shows the verti al omponent of the velo ity and the relatedpressure eld along the line y = 0.2L. In this ase, the numeri al u tuationsof the pressure eld seem larger than the velo ity ones. This results is quitesurprising sin e, in prin iple, the pressure u tuations would lead to a widelyu tuating for e and, therefore, to wide os illations on the velo ity eld. Togive an explanation we onsider the governing equations for a 1D problem

0 0.1 0.2 0.3 0.40.1

0.2

0.3

0.4

0.5

0.6

0.7

x/L

p/ρU2

0 0.1 0.2 0.3 0.4

-0.4

-0.35

-0.3

x/L

v/U

Fig. A.1. 2D jet impinging an orthogonal plate problem: Top: time history of verti alvelo ity omponent along the line y = 0.2L (α = 0.03, ξ = 0.0]) Bottom: time historyof SPH pressure along the line y = 0.2L.23

with no external for e:Dρ

Dt= −ρ ∂u

∂x; ρ

Du

Dt= −∂p

∂x; p = c20(ρ− ρ0) (A.1)If we linearize the previous equations, we obtain the lassi al wave equationfor the velo ity eld u:

∂2u

∂2t= c20

∂2u

∂2x; the related pressure is given by ∂p

∂t= −ρ0 c

20

∂u

∂x(A.2)A possible solution is:

u(x, t) = Ac0 sin[k(x ± c0 t)]

p(x, t) = −Aρ0 c20 sin[k(x ± c0 t)]

(A.3)where the dimensionless parameter A has to be small (A ≪ 1) a ording tothe assumptions made above.Now let's onsider a domain [0, L] where periodi boundary onditions andthe following initial onditions are pres ribed:

u(x, 0) = Ac0 sin4π

Lx

p(x, 0) = −Aρ0 c20 sin

4π

Lx .

(A.4)If we solve this problem with a 1D SPH solver using 200 parti le (with no arti- ial vis osity and no arti ial density diusion) we nd that for A greater than0.002 the solver starts developing instabilities at a time lose to t = 10L/c0.The solution is reported in gure (A.2) where the velo ity and the pressurehave been made dimensioless by dividing respe tively for the sound velo ity c0and for ρ0 c

20. It is evident that using su h a s ale the pressure eld numeri alu tuations have the same amplitude of those presented in the velo ity eld.Now let us onsider the presen e of a large onve tive velo ity eld U0. Thesolution (A.3) be omes:

u(x, t) = Ac0 sin[k(x ± c0 t)] + U0

p(x, t) = −Aρ0 c20 sin[k(x ± c0 t)]

(A.5)In this ase the referen e velo ity U0 and the referen e pressure P0 = ρ0 U20 an modify the per eption of the order of magnitude of the numeri al noise.Indeed if we denote the numeri al u tuations by ∆u and ∆p, we have that

∆u/c0 ≃ ∆p/ρ0c20 (see gure A.3) Conversely, if use U0 and P0 to make theu tuations dimensionless we have ∆u/U0 = ∆p/P0 (U0/c0) and, therefore,24

0.2 0.4 0.6 0.8-0.003

-0.002

-0.001

0

0.001

0.002

0.003

x/L

P/ ρ0 c02

0.2 0.4 0.6 0.8-0.003

-0.002

-0.001

0

0.001

0.002

0.003

u / c0

x/L

t=0

t=10 L / c0

Fig. A.2. SPH solution for the wave equation (A.2) in the domain [0 − L] using asinitial onditions the equations (A.4).

0 0.05 0.1 0.15 0.2

-0.002

-0.001

0

0.001

x/L

[ p - p(x= 0) ]/ρc02

0 0.05 0.1 0.15 0.2

-0.001

0

0.001

0.002

x/L

[v - v(x= 0) ]/c0

Fig. A.3. 2D jet impinging an orthogonal plate problem: Top: time history of verti alvelo ity omponent along the line y = 0.2L (α = 0.03, ξ = 0.0]) made dimensionlessusing the speed of sound c0. Bottom: time history of SPH pressure along the liney = 0.2L made dimensionless with the a ousti pressure ρ0 c2

0.25

∆u/U0 ≪ ∆p/P0 (see gure A.1) as a onsequen e of the weakly ompressibleassumption (4).Referen es[1 J. Bonet, T.S.L. Lok, Variational and momentum preservation aspe ts of SPHformulations, Comput. Meth. Appl. Me h. Engng. 180 (1999) 97115.[2 A. Colagrossi, A meshless Lagrangian method for free-surfa e and interfa eows with fragmentation, PhD thesis, Department of Me hani al Engineering,University of Rome, La Sapienza" (http://padis.uniroma1.it).[3 A. Colagrossi, M. Landrini, Numeri al simulation of interfa ial ows bysmoothed parti le hydrodynami s, J. Comput. Phys. 191 (2003) 448475.[4 G. Coli hio, A. Colagrossi, M. Gre o, M. Landrini, Free-surfa e ow after adam break: a omparative study, Ship Te h. Res. 49 (2002) 95104.[5 G.A. Dilts, Moving least-squares parti le hydrodynami s I. Consisten y andstability, Int. J. Numer. Meth. Engng. 44 (1999) 11151155.[6 G.A. Dilts, Moving least-squares parti le hydrodynami s II. Conservation andboundaries, Int. J. Numer. Meth. Engng. 48 (2000) 15031524.[7 R.A. Gingold, J.J. Monaghan, Smoothed parti le hydrodynami s: theory andappli ation to non spheri al stars, Mon. Not. Roy. Astron. So . 181 (1977) 375389.[8 M. Gre o, A two-dimensional study of green-water loading, PhD Thesis,University of Trondheim (Norway), 2001.[9 J. Hongbin, D. Xin, On riterions for Smoothed Parti le Hydrodynami s kernelsin stable eld, J. Comput. Phys. 202 (2005) 699709.[10 M. Landrini, A. Colagrossi, O.M. Faltinsen, Sloshing in 2-D ows by the SPHmethod, in Pro . 8th Int. Conf. Numer. Ship Hydro., Busan, Korea, 2003.[11 M. Landrini, A. Colagrossi, M. Gre o, M.P. Tulin, Gridless simulationsof splashing pro esses and near-shore bore propagation, Journal of FluidMe hani s, 591 (2007) 183-213.[12 T. Lee, Z. Zhou, Y. Cao, Numeri al simulations of hydrauli jumps in watersloshing and water impa ting. Journal of Fluid Engineering, 124 (2002), 215-226.[13 J.H. Mi hell, On the Theory of Free Stream Lines, Philosophi al Transa tionsof the Royal So iety of London. A, 181 (1890), 389431.[14 D. Molteni, A. Colagrossi, G. Coli hio, On the use of an alternative water stateequation in SPH, Pro . SPHERIC, 2nd International Workshop, UniversidadPolité ni a de Madrid, Spain, May, (2007).26

[15 J.J. Monaghan, R.A. Gingold, Sho k simulation by the parti le method SPH,J. Comput. Phys. 52 (1983) 374389.[16 J.J. Monaghan, Smoothed Parti le Hydrodynami s, Ann. Rev. Astro.Astrophys. 30 (1992) 543574.[17 J.J. Monaghan, Simulating free surfa e ows with SPH, J. Comput. Phys. 110(1994) 399406.[18 J.J. Monaghan, A. Kos, Solitary waves on a Cretan bea h, J. Waterway, Port,Coastal and O ean Engng. 125 (1999) 145154.[19 J.J. Monaghan, SPH without a tensile instability, J. Comput. Phys. 159 (2000)290311.[20 J.J. Monaghan, A. Kos, N. Issa, Fluid motion generated by impa t, J. Waterway,Port, Coastal and O ean Engng. 129 (2003) 250259.[21 J.J. Monaghan, Smoothed Parti le Hydrodynami s, Rep. Prog. Phys. 68 (2005)17031759.[22 J.J. Monaghan, Private Communi ations (2006).[23 J.P. Morris, Analysis of smoothed parti le hydrodynami s with appli ations,PhD Thesis, Monash University (Australia), 1996.[24 S. Sibilla, SPH simulation of lo al s our pro esses, Pro . SPHERIC, 2ndInternational Workshop, Universidad Polité ni a de Madrid, Spain, May, (2007).[25 J.W. Swegle, D.L. Hi ks, S.W. Attaway, Smoothed Parti le Hydrodynami sstability analysis, J. Comput. Phys. 116 (1995) 123134.[26 G. Taylor, Oblique Impa t of a Jet on a Plane Surfa e, Philosophi alTransa tions for the Royal So iety of London. Series A, Mathemati al andPhysi al S ien es, 260 (1966), Issue 1110, 96100.

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